📚 IGCSE OCR Maths Statistics Revision | IGCSE OCR 数学：统计 考点精讲
Statistics is a core component of the IGCSE OCR Mathematics syllabus, testing your ability to collect, represent, analyse and interpret data. This revision guide covers the essential concepts you need to master, from basic averages and charts to probability and cumulative frequency. Understanding these topics will not only help you in the exam but also build valuable skills for handling real-world data.
统计是 IGCSE OCR 数学考试大纲的核心部分，重点考察收集、展示、分析和解读数据的能力。本复习指南涵盖了你需要掌握的所有关键概念，从基本的平均数和图表到概率和累积频数。深入理解这些考点不仅能助你应对考试，也能培养处理现实数据的重要技能。
1. Types of Data | 数据类型
Data can be classified as qualitative or quantitative. Qualitative data describes attributes, such as colours or names, while quantitative data involves numbers. Quantitative data is further divided into discrete data, which can only take specific values (e.g. number of students), and continuous data, which can take any value within a range (e.g. height, weight).
数据可分为定性数据和定量数据。定性数据描述属性，例如颜色或名称；定量数据涉及数字。定量数据又进一步分为离散数据（只能取特定值，如学生人数）和连续数据（在范围内可取任意值，如身高、体重）。
2. Collecting and Organising Data | 数据收集与整理
Data is often gathered through surveys, experiments or observations. Once collected, it is organised into tables, charts or diagrams. Tally charts and frequency tables are common tools to sort raw data before deeper analysis. When designing a survey, avoid biased questions and ensure a representative sample.
数据通常通过调查、实验或观察来收集。收集完成后，需要用表格、图表或图形进行整理。计数表（tally chart）和频数表是在深入分析前对原始数据进行分类的常用工具。设计调查时，要避免诱导性问题，并确保样本具有代表性。
3. Mean, Median, Mode and Range | 平均数、中位数、众数和极差
The three main averages are the mean, median and mode. The mean is calculated by summing all values and dividing by the number of values. The median is the middle value when data is ordered; if there is an even number of values, it is the mean of the two middle numbers. The mode is the most frequent value. The range is the difference between the largest and smallest values, giving a simple measure of spread.
三种主要的平均数是平均数、中位数和众数。平均数是将所有数值相加后除以数值的个数。中位数是将数据排序后的中间值；若有偶数个数据，则取中间两个值的平均数。众数是出现次数最多的值。极差是最大值与最小值之差，用于简单衡量数据的离散程度。
4. Frequency Tables and Averages | 频数表与平均数
When data is presented in a frequency table, the mean is found by multiplying each value by its frequency, summing these products, and then dividing by the total frequency. The median is located by identifying the position (n+1)/2 in the cumulative frequency. The mode remains the value with the highest frequency.
当数据以频数表的形式呈现时，求平均数需要将每个值乘以其频数，将这些乘积相加，再除以总频数。中位数的确定需要根据累积频数找到第 (n+1)/2 个位置。众数仍然是频数最高的那个值。
5. Grouped Data and Estimated Mean | 分组数据与估算平均数
For grouped data, you do not have exact individual values, so you use the midpoint of each class interval as an estimate. The estimated mean is (Σ fx) ÷ (Σ f), where x is the midpoint and f is the frequency. The modal class is the interval with the highest frequency, and the median class can be found using cumulative frequency.
对于分组数据，由于没有精确的单个数值，你需要使用每个组区间的中点作为估算值。估算平均数的公式为 (Σ fx) ÷ (Σ f)，其中 x 是中点，f 是频数。众数所在组是频数最高的区间，而中位数所在组可以通过累积频数来找到。
6. Cumulative Frequency and Quartiles | 累积频数与四分位数
A cumulative frequency table adds up frequencies row by row. Plotting these against the upper class boundaries gives a cumulative frequency curve. From this curve you can read off quartiles: the lower quartile (Q₁) at 25% of total frequency, the median (Q₂) at 50%, and the upper quartile (Q₃) at 75%. The interquartile range (IQR) = Q₃ – Q₁ measures the spread of the middle half of the data.
累积频数表将频数逐行累加。以累积频数为纵轴、组上限为横轴绘制即可得到累积频数曲线。从该曲线上可以读出四分位数：下四分位数（Q₁）在总频数的 25% 处，中位数（Q₂）在 50% 处，上四分位数（Q₃）在 75% 处。四分位距（IQR）= Q₃ – Q₁ 用于衡量中间一半数据的离散程度。
7. Box Plots (Box-and-Whisker Plots) | 箱线图（盒须图）
A box plot displays the minimum, Q₁, median, Q₃ and maximum. The box spans from Q₁ to Q₃ with a line at the median. Whiskers extend to the minimum and maximum values, provided there are no outliers. Outliers are typically defined as values more than 1.5 × IQR below Q₁ or above Q₃. Box plots are useful for comparing distributions.
箱线图展示了最小值、Q₁、中位数、Q₃ 和最大值。箱体从 Q₁ 延伸到 Q₃，并在中位数处画一条线。须线延伸至最小值和最大值（假设没有异常值）。异常值通常定义为低于 Q₁ – 1.5 × IQR 或高于 Q₃ + 1.5 × IQR 的值。箱线图非常适合用来比较数据分布。
8. Scatter Diagrams and Correlation | 散点图与相关
A scatter diagram shows the relationship between two variables. Correlation describes the strength and direction of this relationship: positive (as one increases, the other tends to increase), negative, or zero. A line of best fit can be drawn by eye to model the trend, and it should pass through the mean point of both variables. Interpolation within the data range is reliable, but extrapolation beyond is not.
散点图展示两个变量之间的关系。相关性描述这种关系的强度和方向：正相关（一个增加，另一个也倾向于增加）、负相关或无相关。最佳拟合线可以通过目测画出，用以模拟趋势，并应通过两个变量的均值点。在数据范围内进行内插是可靠的，但外推则不可靠。
9. Probability Basics | 概率基础
Probability measures how likely an event is to happen, expressed as a fraction, decimal or percentage between 0 and 1. Basic rule: P(A) = number of favourable outcomes ÷ total number of outcomes. The sum of probabilities of all possible outcomes is 1. The complement rule states P(not A) = 1 – P(A).
概率衡量事件发生的可能性，用介于 0 和 1 之间的分数、小数或百分数表示。基本规则：P(A) = 有利结果的数量 ÷ 所有可能结果的总数。所有可能结果的概率之和为 1。互补规则：P(非 A) = 1 – P(A)。
10. Tree Diagrams | 树状图
Tree diagrams are used to list outcomes for two or more combined events. Along branches you write probabilities, and at the end you multiply along the branches to find the probability of a specific sequence. If events are independent, probabilities do not change from one branch to the next; if dependent, they adjust based on the outcome. The sum of probabilities on branches from a single point is always 1.
树状图用于列出两个或更多组合事件的结果。沿着分支写出概率，然后将一条路径上的概率相乘，即可得到特定序列发生的概率。如果事件是独立的，概率不会因分支而改变；如果非独立，则需要根据结果进行调整。从同一点出发的所有分支概率之和总是 1。
11. Interpreting Statistical Diagrams | 解读统计图表
Exam questions often provide bar charts, pie charts, histograms or stem-and-leaf diagrams. For pie charts, remember that each sector angle = (frequency ÷ total frequency) × 360°. Histograms for continuous data use area to represent frequency, so frequency density = frequency ÷ class width. Always check labels, scales and units before answering.
考试中经常出现条形图、饼图、直方图或茎叶图。对于饼图，记住每个扇形的角度 = (频数 ÷ 总频数) × 360°。用于连续数据的直方图以面积表示频数，因此频数密度 = 频数 ÷ 组距。在回答问题前，务必检查标签、比例尺和单位。
12. Comparing Data Sets | 数据集比较
When comparing two data sets, use both a measure of central tendency (mean or median) and a measure of spread (range or IQR). The mean is affected by extreme values, whereas the median is not, so choose the appropriate average based on the context. Use comparative language such as ‘on average, higher’ or ‘more consistent, with a smaller spread’.
在比较两组数据时，要同时使用集中趋势的度量（平均数或中位数）和离散程度的度量（极差或四分位距）。平均数受极端值影响，而中位数不受影响，因此要根据上下文选择合适的平均数。使用对比性语言，如“平均更高”或“更一致，离散程度更小”。
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